Choice strength required for the dependence theorem

Determine which fragment of the Axiom of Choice is sufficient to prove the dependence theorem for functions on combinatorial cubes, given that the theorem is not provable in ZF.

Background

The paper proves a dichotomy for every function on a finite product of sets: either the function locally depends on at most one coordinate after a finite partition of each domain set, or there are sequences witnessing a strong dependence on two nonempty groups of coordinates. The proof uses compactness of the Čech–Stone compactification, and hence an instance of the Axiom of Choice.

The authors show that the theorem can fail in models of ZF containing an infinite Dedekind-finite set, while it remains valid in ZF when the domain sets are well-orderable. They explicitly leave unresolved the precise choice-theoretic strength required for the general theorem.